View source for Lemaître–Tolman metric

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In mathematical physics, the '''Lemaître–Tolman metric''' is the spherically symmetric dust solution of [[Einstein's field equations]]. It was first found by [[Georges Lemaître]] in 1933 and [[Richard Tolman]] in 1934 and later investigated by [[Hermann Bondi]] in 1947.  This solution describes a spherical cloud of dust (finite or infinite) that is expanding or collapsing under gravity.  It is also known as the '''Lemaître–Tolman–Bondi metric''' or the '''Tolman metric'''.

==Details==
The metric is:
: <math>\mathrm{d}s^{2} = \mathrm{d}t^2 - \frac{(R')^2}{1 + 2 E} \mathrm{d}r^2 - R^2 \, \mathrm{d}\Omega^2</math>

where:
: <math>\mathrm{d}\Omega^2 = \mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2</math></center>
: <math>R = R(t,r)~,~~~~~~~~ R' = \partial R / \partial r~,~~~~~~~~ E = E(r) > -\frac{1}{2}</math>
The matter is comoving, which means its 4-velocity is:
: <math>u^a = \delta^a_0 = (1, 0, 0, 0)</math>
so the spatial coordinates <math>(r, \theta, \phi)</math> are attached to the particles of dust.

The pressure is zero (hence ''dust''), the density is 
: <math>8 \pi \rho = \frac{2 M'}{R^2 \, R'}</math>
and the evolution equation is
: <math>\dot{R}^2 = \frac{2 M}{R} + 2 E</math>
where
: <math>\dot{R} = \partial R / \partial t</math>

The evolution equation has three solutions, depending on the sign of <math>E</math>,
: <math>E > 0:~~~~~~~~ R = \frac{M}{2 E} (\cosh\eta - 1)~,~~~~~~~~ (\sinh\eta - \eta) = \frac{(2 E)^{3/2} (t - t_B)}{M}~;</math>
: <math>E = 0:~~~~~~~~ R = \left( \frac{9 M (t - t_B)^2}{2} \right)^{1/3}~;</math>
: <math>E < 0:~~~~~~~~ R = \frac{M}{2 |E|} (1 - \cos\eta)~,~~~~~~~~ (\eta - \sin\eta) = \frac{(2 |E|)^{3/2} (t - t_B)}{M}~;</math>
which are known as ''hyperbolic'', ''parabolic'', and ''elliptic'' evolutions respectively.

The meanings of the three arbitrary functions, which depend on <math>r</math> only, are:
* <math>E(r)</math> – both a local geometry parameter, and the energy per unit mass of the dust particles at comoving coordinate radius <math>r</math>,
* <math>M(r)</math> – the gravitational mass within the comoving sphere at radius <math>r</math>,
* <math>t_B(r)</math> – the time of the big bang for worldlines at radius <math>r</math>.

Special cases are the [[Schwarzschild metric]] in [[geodesic coordinate]]s
<math>M =</math> constant (setting <math>2E=-1/(1+r^2)</math> leads to Schwarzschild metric in Novikov coordinates, while setting <math>2E=0</math> leads to Schwarzschild metric in [[Lemaître coordinates]]), and the [[Friedmann–Lemaître–Robertson–Walker metric]], e.g. <math>E = 0~,~~ t_B =</math> constant for the flat case.

==See also==

*[[Lemaître coordinates]]
*[[Introduction to the mathematics of general relativity]]
*[[Stress–energy tensor]]
*[[Metric tensor (general relativity)]]
*[[Relativistic angular momentum]]

== References ==

* {{cite journal | last =Bondi | first =Hermann | authorlink =Hermann Bondi | title =Spherically symmetrical models in general relativity | journal =[[Monthly Notices of the Royal Astronomical Society]] | volume =107 | issue = | pages =410 | publisher = | location = | date =1947 | issn = | doi = 10.1093/mnras/107.5-6.410| id =  | bibcode =1947MNRAS.107..410B }}
* Krasinski, A., ''Inhomogeneous Cosmological Models'', (1997) Cambridge UP, {{ISBN|0-521-48180-5}}
* Lemaître, G., Ann. Soc. Sci. Bruxelles, A53, 51 (1933).
* {{cite journal|last=Tolman |first=Richard C. |authorlink=Richard C. Tolman |title=Effect of Inhomogeneity on Cosmological Models |journal=Proc.  Natl. Acad.  Sci. |volume=20 |issue=3 |pages=169–76 |publisher=National Academy of Sciences of the USA |date=1934 |url=http://www.pnas.org/content/20/3/169.full.pdf |accessdate=2011-01-27 |archiveurl=https://www.webcitation.org/5w35fIoge?url=http://www.pnas.org/content/20/3/169.full.pdf |archivedate=2011-01-27 |pmid=16587869 |pmc=1076370 |url-status=dead }}

{{DEFAULTSORT:Lemaître-Tolman metric}}
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